Optimal. Leaf size=21 \[ \frac {c x \sqrt {\frac {c}{a+b x^2}}}{a} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6720, 191} \[ \frac {c x \sqrt {\frac {c}{a+b x^2}}}{a} \]
Antiderivative was successfully verified.
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Rule 191
Rule 6720
Rubi steps
\begin {align*} \int \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx &=\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx\\ &=\frac {c x \sqrt {\frac {c}{a+b x^2}}}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {c x \sqrt {\frac {c}{a+b x^2}}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 19, normalized size = 0.90 \[ \frac {c x \sqrt {\frac {c}{b x^{2} + a}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 28, normalized size = 1.33 \[ \frac {c^{2} x \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c x^{2} + a c} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.24 \[ \frac {\left (b \,x^{2}+a \right ) \left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {c}{b x^{2} + a}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.74, size = 19, normalized size = 0.90 \[ \frac {c\,x\,\sqrt {\frac {c}{b\,x^2+a}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.46, size = 66, normalized size = 3.14 \[ \begin {cases} c^{\frac {3}{2}} x \left (\frac {1}{a + b x^{2}}\right )^{\frac {3}{2}} + \frac {b c^{\frac {3}{2}} x^{3} \left (\frac {1}{a + b x^{2}}\right )^{\frac {3}{2}}}{a} & \text {for}\: a \neq 0 \\- \frac {c^{\frac {3}{2}} x \left (\frac {1}{b}\right )^{\frac {3}{2}} \left (\frac {1}{x^{2}}\right )^{\frac {3}{2}}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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